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Chapter 1: Problem 12

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$-4 x \geq 10$$

Short Answer

Expert verified
The solution is \(x \leq -2.5\), or \((-\infty, -2.5]\).

Step by step solution

01

Isolate the Variable

To solve the inequality \(-4x \geq 10\), we need to isolate \(x\). Start by dividing both sides by -4. Remember, when dividing both sides of an inequality by a negative number, the inequality sign flips direction.
02

Divide by Negative Value

Perform the division: \(-4x \geq 10\) becomes \(x \leq \frac{10}{-4}\), which simplifies to \(x \leq -2.5\).
03

Write Solution in Interval Notation

The inequality \(x \leq -2.5\) in interval notation is \((-\infty, -2.5]\).
04

Graph the Solution Set

On a number line, graph \((-\infty, -2.5]\). This includes all numbers less than \(-2.5\), and also includes \(-2.5\) itself (solid dot or bracket at \(-2.5\)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Solving
Solving linear inequalities can be thought of as a close cousin to solving linear equations, but with a bit of a twist. The goal remains the same: isolate the variable on one side. In our exercise, we have the inequality \(-4x \geq 10\). This means we want to solve for \(x\) so that any value which fits this inequality is included in our solution.
  • Start by isolating the variable \(x\). This often involves moving terms around using addition or subtraction and then simplifying by multiplication or division.
  • For inequalities, one crucial thing to remember is that when multiplying or dividing both sides by a negative number, the inequality sign flips direction. This is a key concept and makes inequalities unique compared to equalities.
When completing the steps above for our problem, dividing both sides by \(-4\) gives us the inequality \(x \leq -2.5\). This is now our new inequality which narrows down the values \(x\) can take.
Interval Notation
Interval notation is a handy way to describe subsets of the real number line. It tells us exactly where our solutions lie in a compact form. In the case of our solution \(x \leq -2.5\), we're expressing it with interval notation.
  • The closed interval \((a, b]\) includes the end value \(b\), indicated by the square bracket. A round bracket \((\) means the end value is not included.
  • Since our inequality \(x \leq -2.5\) includes the number \(-2.5\), the closed interval \((-\infty, -2.5]\) accurately represents all numbers less than or equal to \(-2.5\).
  • Using infinity \((\pm\infty)\) always comes with a round bracket \((\) because infinity itself is not a number but an idea of endlessness.
Interval notation makes it simple to report large sets of numbers, like every number less than \(-2.5\), with just a small segment of writing.
Graphing Linear Inequalities
Graphing linear inequalities is a visual approach to showcase all potential solutions of an inequality. Through graphing, we get a useful visual representation of the solution set. To graph our solution \((-\infty, -2.5]\), we follow the simple convention:
  • Draw a number line and locate \(-2.5\).
  • Use a solid dot or bracket at \(-2.5\) to indicate it's included in the set. This represents \(x\) can equal \(-2.5\).
  • Shade the number line to the left of \(-2.5\) to capture all values less than \(-2.5\). This represents all the permissible values of \(x\) in this inequality.
This graphical representation clearly communicates the range of numbers that satisfy the inequality \(x \leq -2.5\) and helps in visualizing the limitless range extending to infinity.

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Most popular questions from this chapter

Factor the expression completely. $$18 y^{3} x^{2}-2 x y^{4}$$

Use the Difference of Squares Formula to factor \(17^{2}-16^{2}\). Notice that it is easy to calculate the factored form in your head but not so easy to calculate the original form in this way. Evaluate each expression in your head: (a) \(528^{2}-527^{2}\) (b) \(122^{2}-120^{2}\) (c) \(1020^{2}-1010^{2}\) Now use the Special Product Formula \((A+B)(A-B)=A^{2}-B^{2}\) to evaluate these products in your head: (d) \(79 \cdot 51\) (e) \(998 \cdot 1002\)

Multiply the algebraic expressions using a Special Product Formula and simplify. $$(y+2)^{3}$$

Factor the expression completely. $$12 x^{3}+18 x$$

Perform the indicated operations and simplify. $$x^{1 / 4}\left(2 x^{3 / 4}-x^{1 / 4}\right)$$
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